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Finding dependencies between frequencies with the kernel cross-spectral density

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Besserve,  M
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Department Physiology of Cognitive Processes, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Janzing,  D
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Logothetis,  NK
Department Physiology of Cognitive Processes, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Schölkopf,  B
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Besserve, M., Janzing, D., Logothetis, N., & Schölkopf, B. (2011). Finding dependencies between frequencies with the kernel cross-spectral density. In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2011) (pp. 2080-2083). Piscataway, NJ, USA: IEEE.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-BBCA-6
Abstract
Cross-spectral density (CSD), is widely used to find linear dependency between two real or complex valued time series. We define a non-linear extension of this measure by mapping the time series into two Reproducing Kernel Hilbert Spaces. The dependency is quantified by the Hilbert Schmidt norm of a cross-spectral density operator between these two spaces. We prove that, by choosing a characteristic kernel for the mapping, this quantity detects any pairwise dependency between the time series. Then we provide a fast estimator for the Hilbert-Schmidt norm based on the Fast Fourier Trans form. We demonstrate the interest of this approach to quantify non-linear dependencies between frequency bands of simulated signals and intra-cortical neural recordings.